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Question

A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The coefficients of linear expansion of the two metals are αc and αb. On heating, the temperature of the strip goes up by ΔT and the strip bends to form an arc of radius of curvature R. Then R is,
[Neglect the change in width of the rods due to expansion]

A
Proportional to ΔT
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B
Inversely proportional to ΔT
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C
Proportional to |αbαc|
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D
Inversely proportional to |αbαc|
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Solution

The correct option is B Inversely proportional to ΔT
Let w and l0 be width and initial length of the two strips, respectively.
The length after heating becomes,
lc=l0(1+αcΔT)
lb=l0(1+αbΔT)
The strips bend as shown in the figure.

From the figure, the length of two strips are,
lc=Rθ
lb=(R+w)θ
Dividing the above two equation
R+wR=lblc=1+αbΔT1+αcΔT

=1+αbΔT)(1+αcΔT)1

=(1+αbΔT)(1αcΔT) [From binomial expansion]

=1+(αbαc)ΔTαbαcΔT2

αbαcΔT2 is a very small quantity; so we can ignore it.

So, R+wR1+(αbαc)ΔT

1+wR=1+(αbαc)ΔT

R=w(αbαc)ΔT

Radius is inversly proportional to ΔT.

Hence, (B) is the correct answer.

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